Note 2

The Twenty Most Important Discoveries in Mathematics

If one were to leave the discipline of mathematics, it would profit them to leave it remembering only twenty-five discoveries. All else may be forgotten, or left behind in books for future reference, but the following should be something every educated person commits to memory. They are quite simply the foundation stone upon which all mathematics (as well as our physical universe) rests, and perhaps the latter is of prime importance for us.

1. Gödel’s Incompleteness Theorem

A theory, in mathematics, is complete when every statement or its opposite is provable from within the theoryand consistent when it contains no contradictions. In the early 1930s, the formalism movement ruled the mathematics world. Formalism intended to verify the completeness and consistency of each theory and to algorithmically resolve whether any given statement fit the theory.

However, in 1931 Austrian-born logician Kurt Gödel published this theorem and it has had an important impact on modern mathematics. His two-part proof stated that in any strictly logical mathematical theory there are propositions (or statements) that can never be proven or disproven based on the axioms within that system. He initially proved that any consistent mathematical theory comprising the natural numbers is incomplete. Secondly, he proved that any such theory cannot include a proof of its own consistency; consistency requires some larger theory still, but proving consistency within this larger theory requires an even larger one. His theorem preordained that the formalists’ goals of creating completeness and consistency would fail for any theory comprising the natural numbers, and terminated any further efforts to do so.

2. The Fundamental Theorem of Arithmetic

Any number greater than 1 is either a prime number (a number divisible by only 1 and itself) or the unique product of prime numbers; therefore, prime numbers are the “building blocks” of all other numbers.

3. The Fundamental Laws of Algebra

a. Commutative law of addition, a + b = b + a: regardless of the order one carries them out the sum is identical. For example, 4 + 6 = 6 + 4 = 10, and (-4) + (-6) = (-6) + (-4) = -10. Mathematicians call this law commutative since the numbers switch places without altering the total.

The decimal method of counting is ancient for scholars ascribe a knowledge of place values to the Mesopotamians and Egyptians of about 3400 B.C.E. Probably shortly thereafter, the Mohenjo-Daro culture in the Indus Valley begins using a form of the decimal system. India knew it in long before its appearance in the writings of the Arabs; China adopted it from Buddhist missionaries; and Muhammad Ibn Musa al-Khwarizmi, the greatest mathematician of his age (d. 850?), seems to have introduced it into Baghdad.

Probably tied to the creation of the decimal system, or the cause of its creation, was the abacus. The earliest methods (5th millennium B.C.E.) used by the Mesopotamians to calculate involved using a board covered with sand or dust divided into columns of ten. The word “abacus” etymologically comes from either the Hebrew abaq, denoting to “wipe the dust” or the Greek abax, “board covered with dust.” Perhaps both words are cognate.

The oldest known use of zero in the Old World is in an Arabic document dated to 873, three years sooner than its first known appearance in India. However, by general consent the Arabs borrowed this too from India where it may have been created around 200 B.C.E. prior to its discovery by the Mayas: therefore, the most modest and most valuable of all numerals is one of the subtle gifts of India to humankind.

5. Invention of Geometry

The dependence of Egyptian life upon the fluctuations of the Nile led to careful records and calculations of the rise and recession of the river; surveyors and scribes were continually re-measuring the land whose boundaries the inundation had obliterated, and this measuring of the land was evidently the origin of geometry. Nearly all the ancients agreed in ascribing the invention of this science to the Egyptians, though probably other civilizations developed it simultaneously.

6. The Impossibility of Trisecting the Angle and Doubling the Cube

These two famous problems required the construction of a particular line or figure by utilizing only a straightedge and compass, but both defied the endeavors of countless generations of ancient Greek mathematicians to resolve. Later mathematicians concocted many attempts particularly in the area of non-rational numbers. In the late nineteenth century, they finally resolved what a compass and straightedge were capable of doing, and determined that these two problems are beyond the scope of both instruments.

7. The Denumerability of the Rational Numbers

In the late nineteenth century, German mathematician Georg Cantor invented set theory and proved that there are different degrees of infinity—actually an infinite quantity of them. He made famous numerous paradoxical consequences of this and had an overwhelming impact on the ensuing development of the field.

A number is connected with all sets containing that many elements. For any two of these sets, a 1-to-1 equivalence exists amongst the elements of each set. This number is their cardinality. But assume the sets are infinite. For instance, consider every natural number, 1, 2, 3 . . . as one set, and every even number 2, 4, 6 . . . as the other. By linking any number in set one with its double in set two, we have a perfect 1-to-1 correspondence. Via Cantor’s contention, the two sets are identical in size. This is paradoxical!

Cantor took these ideas considerably further, confirming specifically that the set of all real numbers has a degree of infinity (cardinality) larger than the natural numbers. He accomplished this by applying a resourceful method called the “diagonal argument,” and worked out a whole hierarchy of transfinite cardinal numbers, the least of which is the cardinality of the natural numbers known as “d.”

8. Pi (π) ≈ 3.1415926 . . . .

Pi, the ratio of the circumference of a circle to its diameter, is a universal constant for all circles. It is an irrational number signifying that it is not the ratio of two integers.

Several important mathematical and physical equations include π. It appears often in problems of some periodic phenomena (e.g. the motion of pendulums or alternating electric currents), in the segments of arcs or other curves, the areas of ellipses, sectors, and additional curved surfaces, and in the volumes of numerous solids.

In 1882, German mathematician Ferdinand Lindemann established that π is a transcendental number—i.e. it is not the root of any polynomial equation with rational coefficients. He demonstrated that it is impossible to “square the circle” i.e. construct a square whose area equals that of a given circle (a problem that had vexed ancient Greek mathematicians) using algebra or a ruler and compass since the area of a square can always be stated as a polynomial equation with rational coefficients.

9. F = ma

Sir Isaac Newton discovered this his second law of motion in the mid-1660s and published it in 1687. It is among the most critical laws in all of physics. It states that bodies do not speed up, slow down, or change direction without being pushed in some way, and that an acceleration (i.e. a change in velocity in the direction of the force) ensues. If a fixed force acts on two bodies of dissimilar masses, the body with smaller mass will have a greater acceleration. Physicists measure force in units of newtons (N), where 1 N accelerates 1 kg at the rate of 1 m/s2.

Newton’s second law likewise pertains to bodies moving in a circle. If force applies at a right angle, a body will turn. If the force continuous, the body will circle. This is centripetal force, and it constantly points to the center of the circle. If the force ceases, the body moves in a straight line in the direction it was heading when the force stopped.

10. E = mc²

In 1905, Albert Einstein published his special theory of relativity. In September of that year, he published a paper where he examined a consequence of the theory—energy and mass are two sides of the same coin. Since the speed of light squared (c2) is colossal, the energy of a given mass is enormous. For example, 1 kg of matter entirely converted into energy would equal the explosion of 19.487 megatonnes of TNT; said another way, a mass of one gram generates about 25 million kilowatt-hours of energy.

When Einstein published this, scientists were unaware of any mechanism that would change mass into energy or energy into mass. They currently know of two ways to do the former: fusion and fission. In both, the atoms at the end of each process have less mass than they did at the start.

11. Schrödinger’s Equation (iћ ∙ ∂/∂t – ψ = Ĥψ)

In 1925, Austrian physicist Erwin Schrödinger fused the theoretical work of Max Planck, Niels Bohr, Louis de Broglie, and Werner Heisenberg into what became the foundational equation of quantum mechanics. It depicts electrons and other subatomic particles that compose matter as waves—but waves with an odd interpretation: the wave acts to localize the electron, but the electron must always be a whole particle. The equation regards subatomic particles as occurring only in specified permissible energy states whose position and momentum are portrayed in terms of probability and not certainty—a radical departure from classical physics.

The equation has numerous solutions, but only those that are finite and have only one value are useful. These solutions link to values (aka eigenvalues) of the electron’s energy level. Schrödinger solved Danish physicist Niels Bohr’s equation for the hydrogen atom* and the eigenvalues of the electron’s energy agreed with those of the energy levels given by the older theory. However, the solutions are wave functions, and Max Born demonstrated that these wave functions would signify only probabilities. To be more explicit, the square of the height of the wave at a given point in space gives the probability that an electron is at that spot. It is probably where Bohr predicted it would be, but it does not travel in a precisely circular orbit. Scientists have come to portray its location by the more complex concept of an orbital—or area in space where the electron has a fluctuating degree of probability of being located. In the “real” world, the observer seems to determine which one wave value is true. Scientists refer to this as “wave collapse” and it gives rise to the famous “Schrödinger’s cat” thought experiment.

* This portrayed electrons revolving around atoms in precise circular orbits at particular distances that corresponded to specific energy levels.

12. Second Law of Thermodynamics (dS≥0)

Nature appears to “favor” disorder aka entropy (S). More precisely entropy measures the nearness a system is to equilibrium. Consequently, when an isolated system attains full entropy, it can’t experience further change. S can either remain unchanged or rise but never decline on its own. A closed system’s total entropy rises as the system does work. When it attains maximum entropy, it is unable to do useful work, and without work heat cannot move from an area of lower temperature to one of higher temperature.

A “perpetual-motion machine” is a machine that produces work while defying the second law because energy would constantly be extracted from a cold location to do work in a hot location without cost. Scientists occasionally define the second law as an assertion that precludes this. Since heat has the highest disorder of any kind of energy, heat engines†can convert some but not all of the heat accessible to them into mechanical energy;the remainder just heats the environment. Conversely, mechanical energy can totally transform to heat. In each conversion, the total quantity of energy stays conserved, obeying the first law of thermodynamics, but the second law defines a constraint on the direction that the conversion of energy can take place.

The basic laws of physics do not necessitate that time move in a specific direction; the identical physical laws hold true whether time moves backward or forward. However, the second law provides an important “arrow” to the direction in which time flows because the entropy of a closed system must increase over time.

† Machines that generate mechanical energy while transferring heat into their environments.

Every chemistry student is familiar with the ideal gas law PV = nRT, where P signifies the pressure of the gas, V the volume, n the number of molecules, T the temperature, and R the “universal gas constant.” However, not all of them are familiar with the formulation derived from statistical mechanics where the equation is rewritten as PV = NkT. Here k, named for Ludwig Boltzmann, is an underlying constant that correlates the average kinetic energy of the constituents of the gas to the temperature of that gas. As rewritten P is the absolute pressure in pascals, N the number of molecules per volume of gas, and T the absolute temperature. Boltzmann’s constant played a major role in Planck’s development of quantum physics.

James Clerk Maxwell synopsized all known facts of electric and magnetic phenomena with these four equations that elegantly explain the creation and interaction of electric and magnetic fields.

From his equations, Maxwell made a prediction that a changing electric field would induce a magnetic field and vice versa. This led him to propose the existence of electromagnetic waves, and he predicted that scientists could generate them in the laboratory. After calculating an electromagnetic wave’s velocity, he found it to be identical to that of light waves and determined that they are the same phenomena. Once thought a fundamental law of the universe, these equations do not pertain to events governed by quantum theory, wave mechanics, or relativity.

15. Universal Law of Gravitation Fg = G ∙ (M1 ∙ M2)/r2

“Newton’s law of gravitation,” writes Ian Stewart, “synthesized, in one simple mathematical formula, millennia of astronomical observations and theories. It explained many puzzling features of planetary motion, and made it possible to predict the future movements of the Solar System with great accuracy. Einstein’s theory of general relativity eventually superseded the Newtonian theory of gravity . . . but for almost all practical purposes the simpler Newtonian approach still reigns supreme.

. . . This law demonstrated the enormous power of mathematics to find hidden patterns in nature and to reveal hidden simplicities behind the world’s complexities.”

16. The Normal Distribution y = 1/[σ ∙ (2π)] ∙ e-(x-μ) ²/2σ²

“The probability of observing a particular data value,” writes Ian Stewart, “is greatest near the mean value—the average—and dies away rapidly as the difference from the mean increases. How rapidly depends on a quantity called the standard deviation [σ].

. . . [This equation] defines a special family of bell-shaped probability distributions, which are often good models of common real-world observations.

What does it lead to? The concept of the ‘average man,’ tests of significance of experimental results, such as medical trials, and an unfortunate tendency to default to the bell curve as if nothing else existed.”

17. The Golden Ratio φ ≈ 1.618. . . .

“The Golden ratio,” writes Elaine J. Hom “is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi [φ]. . . . In an equation form, it looks like this:

a/b = (a+b)/a = 1.6180339887498948420 . . .

. . . Around 1200, mathematician Leonardo Fibonacci discovered the unique properties of the Fibonacci sequence. This sequence ties directly into the Golden ratio because if you take any two successive Fibonacci numbers, their ratio is very close to the Golden ratio. As the numbers get higher, the ratio becomes even closer to 1.618.

. . . Phi . . . appears around us in our daily lives, even in our aesthetic views. Studies have shown that . . . [f]aces judged as the most attractive show Golden ratio proportions between the width of the face and the width of the eyes, nose, and eyebrows. The test subjects weren’t mathematicians or physicists familiar with phi—they were just average people, and the Golden ratio elicited an instinctual reaction.

Among the most vital parts of our Oriental heritage are the “Arabic” numerals and the decimal system, both of which came to us, through the Arabs, from India. The miscalled “Arabic” numerals are found on the Rock Edicts of Ashoka (256 B.C.E.), a thousand years before their occurrence in Arabic literature.

In 773, at the caliph al-Mansur’s behest, translations were made of the Siddhamas—Indian astronomical treatises dating as far back as 425 B.C.E.; these versions may have been the vehicle through which the Arabic numerals and the zero were brought from India into Islam. In 813 Muhammad ibn Musa al-Khwarizmi (780-850?)—perhaps the greatest of medieval mathematicians—used the Hindu numerals in his astronomical tables; about 825 he issued a treatise known in its Latin form as Algoritmi de numero Indorum—“al-Khwarizmi on the Numerals of the Indians”; in time algorithm came to mean any arithmetical system based on the decimal notation.

19. Imaginary Numbers: i² = -1

The “real numbers” are numbers found on a number line.Imaginary numbers are numbers written as a real number multiplied by i; an example of i would be the square root of -1. Any imaginary number squared is a negative number. How did this come about? Mathematicians wanted the ability to obtain square roots of negative numbers and couldn’t if they limited themselves to real numbers, so they created i. Taking a real number such as 9 and multiplying it by i creates an imaginary number 9i.

20. Euler’s Formula for Polyhedrons: F – E + V = 2

“The numbers of faces, edges, and vertices of a solid,” writes Ian Stewart, “are not independent, but are related in a simple manner.

. . . [This equation] distinguishes between solids of different topologies using the earliest example of a topological invariant. This paved the way to more general and more powerful techniques, creating . . . one of the most important and powerful areas of pure mathematics: topology, which studies geometric properties that are unchanged by continuous deformations. Examples include surfaces, knots, and links. Most applications are indirect, but its influence behind the scenes is vital. It helps us understand how enzymes act on DNA in a cell, and why the motion of celestial bodies can be chaotic.”